Solve The Equation For Y Y Y To Find The Inverse. X = Y 6 + 12 X = \frac{y}{6} + 12 X = 6 Y ​ + 12

Introduction

In mathematics, an inverse function is a function that reverses the operation of another function. To find the inverse of a function, we need to solve the equation for the variable that is not the input. In this article, we will solve the equation for $y$ to find the inverse of the given function $x = \frac{y}{6} + 12$.

Understanding the Function

The given function is $x = \frac{y}{6} + 12$. This is a linear function, and it can be written in the slope-intercept form as $x = \frac{1}{6}y + 12$. The slope of this function is $\frac{1}{6}$, and the y-intercept is 12.

Step 1: Write the Equation in Terms of $y$

To find the inverse of the function, we need to write the equation in terms of $y$. We can do this by isolating $y$ on one side of the equation.

x = \frac{y}{6} + 12

Step 2: Subtract 12 from Both Sides

To isolate $y$, we need to subtract 12 from both sides of the equation.

x - 12 = \frac{y}{6}

Step 3: Multiply Both Sides by 6

To get rid of the fraction, we can multiply both sides of the equation by 6.

6(x - 12) = y

Step 4: Simplify the Equation

Now, we can simplify the equation by distributing the 6 to the terms inside the parentheses.

6x - 72 = y

Step 5: Write the Equation in Terms of $y$

Now, we have the equation in terms of $y$. We can write it as:

y = 6x - 72

Conclusion

In this article, we solved the equation for $y$ to find the inverse of the given function $x = \frac{y}{6} + 12$. We followed the steps to isolate $y$ and wrote the equation in terms of $y$. The inverse function is $y = 6x - 72$.

Understanding the Inverse Function

The inverse function $y = 6x - 72$ is a linear function with a slope of 6 and a y-intercept of -72. This function reverses the operation of the original function $x = \frac{y}{6} + 12$. To find the inverse of a function, we need to solve the equation for the variable that is not the input.

Real-World Applications

The concept of inverse functions has many real-world applications. For example, in physics, the inverse of the velocity function is used to find the acceleration of an object. In economics, the inverse of the demand function is used to find the supply function.

Common Mistakes

When solving the equation for $y$ to find the inverse, there are some common mistakes to avoid. One of the mistakes is to forget to isolate $y$ on one side of the equation. Another mistake is to not simplify the equation after multiplying both sides by 6.

Tips and Tricks

To solve the equation for $y$ to find the inverse, here are some tips and tricks:

• Make sure to isolate $y$ on one side of the equation.
• Simplify the equation after multiplying both sides by 6.
• Use the inverse function to reverse the operation of the original function.
• Use the inverse function to find the acceleration of an object in physics.
• Use the inverse function to find the supply function in economics.

Conclusion

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Introduction

In our previous article, we solved the equation for $y$ to find the inverse of the given function $x = \frac{y}{6} + 12$. In this article, we will answer some frequently asked questions about solving the equation for inverse.

Q: What is the inverse of a function?

A: The inverse of a function is a function that reverses the operation of another function. To find the inverse of a function, we need to solve the equation for the variable that is not the input.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, we need to follow these steps:

1. Write the equation in terms of $y$.
2. Isolate $y$ on one side of the equation.
3. Simplify the equation after multiplying both sides by 6.

Q: What is the difference between the original function and its inverse?

A: The original function and its inverse are two different functions that have the same input and output values. The original function takes the input value and produces the output value, while the inverse function takes the output value and produces the input value.

Q: Can I use the inverse function to solve problems in physics and economics?

A: Yes, you can use the inverse function to solve problems in physics and economics. For example, in physics, the inverse of the velocity function is used to find the acceleration of an object. In economics, the inverse of the demand function is used to find the supply function.

Q: What are some common mistakes to avoid when solving the equation for inverse?

A: Some common mistakes to avoid when solving the equation for inverse include:

• Forgetting to isolate $y$ on one side of the equation.
• Not simplifying the equation after multiplying both sides by 6.
• Using the wrong variable as the input or output.

Q: How do I know if I have found the correct inverse function?

A: To know if you have found the correct inverse function, you need to check if the inverse function satisfies the following conditions:

• The inverse function must be a function that takes the output value of the original function and produces the input value.
• The inverse function must be a one-to-one function, meaning that each input value produces a unique output value.
• The inverse function must be a continuous function, meaning that it has no gaps or jumps.

Q: Can I use the inverse function to solve problems in other fields?

A: Yes, you can use the inverse function to solve problems in other fields, such as engineering, computer science, and data analysis. The inverse function is a powerful tool that can be used to solve a wide range of problems.

Q: How do I apply the inverse function to real-world problems?

A: To apply the inverse function to real-world problems, you need to follow these steps:

1. Identify the problem and the variables involved.
2. Write the equation in terms of $y$.
3. Isolate $y$ on one side of the equation.
4. Simplify the equation after multiplying both sides by 6.
5. Use the inverse function to solve the problem.

Conclusion

In conclusion, solving the equation for inverse is an important concept in mathematics that has many real-world applications. By following the steps and avoiding common mistakes, we can find the inverse of a function and use it to solve problems in physics, economics, and other fields.